ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001 
375 
operator is invariable in different directions, it means you can 
get the same detected edges when you rotate the operator 
The Laplace operator not only response to image edges, but 
also to corner points, end points of a line and isolated points. 
To restrain noise, the LOG operator smoothes an image at 
first, then performs the difference. Though the image noise is 
partly restrained, the detected result is influenced. From 
above discussion, we can find the operators which are relate 
to direction are not effective to detecting edges when the 
image is complicated and there are abundant edges in the 
image. In addition, Due to noise in an image the detected 
result is not perfect if the operators are directly used. 
Therefore in this paper, Our objective is to introduce a new 
method to detect image edge so as to get a better result. 
2.The Á Trous Wavelet Decomposition 
2.1Wave!et Decomposition 
Wavelet analysis has been successfully used in image 
processing field. Wavelet transform is a new method which 
can decompose an image into different resolution images. 
Suppose function ijj (x) e L 2 (R)(L 2 (R) is quadratically 
integrable space), and ijj(x) satisfies: 
|2 
+ r v(e>)‘ 
C r = 2 n J—j—r—dco +co 
(1) 
or \i//{x)dx = 0 
(2) 
Where ijj(u)) is the Fourier transform result of ijj(x). 
When ijj(x) satisfies equation (1 ) or ( 2 ) and can rapidly 
converge, ip(x) is called basic wavelet. 
ijj(x) flexes a and shifts b then we have: 
VaA X ) 
I 4 (x-b 
M 2 v / — 
^ a 
(3) 
Where a, beR, a*0 
and ijja.b(x) is called wavelet. 
Wavelet transform of distribution function f(x) can be defined as 
following 
+°° i r 1 
wf(a,b)= ^f(x)-\a\iy/\a A (x-b)\lx (4) 
can’t be used in computation by program. To do it by computer, 
we must make the continuous wavelet transform discrete. There 
are many methods to perform discrete wavelet transform, such 
as pyramidal algorithm which make use of orthogonal basis to 
decompose an image (or a signal), but the dimension of the result 
image is changed, that is not advantageous in some applications 
like pattern recognition and image fusion etc. 
To get the result image of the same dimension as the original one, 
we adopt the algorithm: ä trous algorithm. The discrete approach 
of the wavelet transform can be done with the special version of 
the so-called ä trous algorithm (with holes). The algorithm can 
decompose an image (or a signal) into an approximate signal and 
a detail signal at a scale, the detail signal is called a wavelet 
plane, which is the same as the original image in dimension. 
Suppose that the sampled image data C 0 (k) are the scalar 
products at pixels k of the function f(x) with a scaling function cj)(x) 
which corresponds to a low pass filter. The first filtering is then 
performed by a twice magnified scale leading to the C,(k) set. 
The image difference C 0 (k)-Ci(k) which is called first wavelet 
plane contains the information between those two scales and is 
the discrete set associated with the wavelet transform 
corresponding to 4>(x). The associated wavelet is therefore ijj(x). 
(5) 
The distance between samples increasing by a factor 2 from the 
scale (i-1) (¡>0) to the next one, Ci(x) is given by 
C,(*) = 2>(/)-C M (* + 2 w /) (6) 
and the discrete wavelet transform w¡(x) by: 
w,(x) = C l _ l (x)-C l (x) 
(7) 
Where Wi(x) is the wavelet coefficient and Cj(x) is approximate 
signal at the i scale, h(l) is a low pass filter. 
The coefficients h(l) is derived from the scaling function <£(x): 
l \l) i 
(8) 
The algorithm can be used to rebuild the data frame because the 
last smoothed array C np is added to all the differences w ( 
up 
c 0 (x) = c„ p w+X w J W 
7=1 
(9) 
Where f(x) eL 2 (R),beR, y/ (x)is complex conjugate of ip(x). 
2.2 The A Trous Wavelet Decomposition Method 
Continuous wavelet transform definition is introduced above. It If the linear interpolation for the scaling function i>(x) (see figure